Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are ` hat i+2 hat j- hat k` and `- hat i+ hat j+ hat k` respectively, in the ratio 2 : 1 (i) internally (ii) externally

Explanation

(i)`\vec{O P}=\hat{i}+2 \hat{j}-\hat{k}`

`\vec{OQ} =-\hat{i}+\hat{j}+\hat{k}`

`\vec{O R}=\frac{2 \times \vec{OQ}+1 . \vec{O P}}{2+1}`
`=\frac{2(-\hat{i}+\hat{j}+\hat{k})+(\hat{i}+2 \hat{j}-\hat{k})}{3}`


`\vec{O R}=\frac{-\hat{i}+4 \hat{j}+\hat{k}}{3}`

(ii)`\vec{O R} =\frac{2 \times \vec{OQ}-1 \times vec{OP}}{2-1}`

`=\frac{2(-\hat{i}+\hat{j}+\hat{k})-(\hat{i}+2 \hat{j}-\hat{k})}{2-1}`

` \vec{O R} =\frac{-3 \hat{l}+3 \hat{k}}{1}`